Splitting field of cyclotomic polynomials over $\mathbb{F}_2$.

Question

Let $\Phi_5$ be the 5th cyclotomic polynomial and $\Phi_7$ the 7th. These polynomials are defined like this: $$ \Phi_n(X) = \prod_{\zeta\in\mathbb{C}^\ast:\ \text{order}(\zeta)=n} (X-\zeta)\qquad\in\mathbb{Z}[X] $$ I want to calculate the splitting field of $\Phi_5$ and the splitting field of $\Phi_7$ over $\mathbb{F}_2$. In $\mathbb{F}_2[X]$ we have $$ \Phi_5(X) = X^4 + X^3 + X^2+X+1 $$ and $$ \Phi_7(X) = (X^3+X+1)(X^3+X^2+1) $$ My question is: what are the splitting fields of the polynomials? I already know it should be of the form $\mathbb{F}_{2^k}$ for some $k\in\mathbb N$. Also the degree of every irreducible factor of a cyclotomic polynomial in $\mathbb{F}_q[X]$ is equal to the order of $q\in(\mathbb{Z}/n\ \mathbb{Z})^\ast$, assuming $(q,n)=1$.

Answer 1

Since we want the degree of an irreducible factor to be equal to one, we want $$ \text{order} (2^k) =1 $$ in $(\mathbb{Z} / 5\mathbb{Z})^\ast$. The only element with this order is 1. Therefore we search the smallest $k$ such that $2^k\equiv 1\mod 5$. A bit puzzzling gives us $$ 2^1=2\\ 2^2=4\\ 2^3=8=3\\ 2^4=16=1. $$ Therefore the splitting field of $\Phi_5$ should be $\mathbb{F}_{2^4}$.

Is this correct?

Answer 2

Hint :

• If $f(X)$ is irreducible in $F[X]$ then $F[X]/(f(X))$ is a field.
• any polynomial $f(X)$ has root in $F[X]/(f(X))$ (which need not be a field in general)
• What is the cardinality of $\mathbb{F}_2[X]/(X^4 + X^3 + X^2+X+1)$.
• How many finite fields of cardinality $n$ can you list out for a given $n$.
• Splitting field of $f(X)g(X)$ is contains splitting field of $f(X)$
• Splitting field of $(X^3+X+1)(X^3+X^2+1)$ is contains splitting field of $(X^3+X^2+1)$
• As $(X^3+X^2+1)$ is irreducible in $\mathbb{F}_2[X]$ its splitting field would be (???)
• What is splitting field of $(X^3+X+1)$.
• Do you see some relation between splitting field of $(X^3+X+1)$ and of $(X^3+X^2+1)$.

Can you now conclude??